群作用的无定形复杂性及其在准晶中的应用
Amorphic complexity of group actions with applications to quasicrystals
摘要：
本文定义了紧度量空间上局部紧$σ$ -紧顺从群作用的无定形复杂性。无定形复杂性最初是为$\mathbb Z$ -作用引入的，是一个拓扑不变量，它度量了动力系统在零熵状态下的复杂性。我们表明，它是量身定制的研究严格遍历群作用与离散谱和连续的本征函数。这类行动包括，特别是，Delone动力系统相关的定期模型集通过迈耶的削减和项目的方法。我们提供了尖锐的上界的无定形复杂性，这样的系统。在这样做的时候，我们观察到无定形复杂性和分形几何之间的密切关系。
英文摘要：
In this article, we define amorphic complexity for actions of locally compact $σ$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb Z$-actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via Meyer's cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.
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